Math 712. Topics in Mathematical Physics: Energy estimates for the wave equation on curved background.
- Fall 22 - Hans Lindblad

The course will be about local energy decay estimates for the wave equation on curved background with applications to the wave equation on black hole background in general relativity. For small perturbations of flat Minkowski space time one has local energy decay estimates, due to dispersion of waves. However for larger perturbations there is the possibility of stationary or time periodic solutions that do not decay, as well as waves concentrated on rotating trapped geodesics, that has to be ruled out.

The stability of black holes with small angular momentum was recently proven, but the case of large angular momentum requires different methods. The intent of the course is to go over approaches to study this problem in the simpler situation of the wave equation on the background of a metric. The course does not require any previous exposure to general relativity but a basic knowledge of the energy estimate and Fourier transform methods for the wave equation as well as the spectral theorem, as in the graduate PDE sequence would be helpful. 

We will start with following Metcalfe, Sterbenz and Tataru Local energy decay for scalar fields on time dependent nontrapping backgrounds which deals with the case close to flat space. Then we will study : Dafermos, Rodnianski, Shlapentokh-Rothman Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M.